Hilbert's third problem

Given any two polyhedra of equal volume in three-dimensional Euclidean space $\mathbb{R}^3$, is it always possible to cut the first into finitely many polyhedral pieces and reassemble them by rigid motions to form the second? Hilbert conjectured that the answer is no, and asked for two polyhedra of equal volume that are not scissors-congruent.

Max Dehn, a student of Hilbert, solved the problem in 1900 — making it the first of Hilbert's 23 problems to be resolved — by introducing the Dehn invariant $\operatorname{Dehn}(P) = \sum_e \ell(e) \otimes [\theta(e)] \in \mathbb{R} \otimes_{\mathbb{Q}} (\mathbb{R}/\pi\mathbb{Q})$, where $\ell(e)$ is the length and $\theta(e)$ the dihedral angle of each edge $e$. Any finite polyhedral dissection preserves $\operatorname{Dehn}(P)$; a cube has $\operatorname{Dehn} = 0$ because its dihedral angles are $\pi/2$, whereas a regular tetrahedron has $\theta = \arccos(1/3)$, which is not a rational multiple of $\pi$, giving $\operatorname{Dehn} \ne 0$. Dehn's paper appeared in Mathematische Annalen in 1901.